{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T07:48:46Z","timestamp":1774856926893,"version":"3.50.1"},"reference-count":53,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03","funder":[{"name":"Henan Soft Science Research Program Project","award":["No. 252400410555"],"award-info":[{"award-number":["No. 252400410555"]}]},{"name":"Universities of Henan Province","award":["No. SKJYB2023-05"],"award-info":[{"award-number":["No. SKJYB2023-05"]}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["No. 72501132"],"award-info":[{"award-number":["No. 72501132"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Unc. Fuzz. Knowl. Based Syst."],"published-print":{"date-parts":[[2026,5]]},"abstract":"<jats:p>Uncertainty widely exists in standby redundancy system designing, and it is of great significance for decision makers in acquiring the system with less cost but longer lifetime. This paper explicitly integrates the uncertainties from the lifetime and cost of the units and the resource limitations in cost and weight, in an expected lifetime-maximization standby redundancy system optimization problem. To deal with the associated uncertain mathematical model, we first utilize the uncertain chance constrained programming to manage the uncertainties on cost constraints. Then, by identifying the weakness of the 99-method, the traditional uncertain expected value simulation method, on the simulation accuracy and efficiency, this paper further proposes ordinary integration method (OIM), and Gauss-Legendre quadrature method (GLQM) based uncertain expected simulations for the objective function of lifetime. The experiment results show that GLQM defeats the others no matter on the simulation accuracy or the efficiency, and it is worth mentioning that it can return accurate results in the case that the inverse uncertainty distribution of the uncertain parameters is linear and the objective function is increasing. Finally, we propose a two-phase approach by integrating GLQM into GA to solve the mathematical model, and further verify the necessity of the uncertainty consideration through a series of numerical examples.<\/jats:p>","DOI":"10.1142\/s021848852650011x","type":"journal-article","created":{"date-parts":[[2026,3,30]],"date-time":"2026-03-30T06:45:52Z","timestamp":1774853152000},"page":"253-280","source":"Crossref","is-referenced-by-count":0,"title":["Gauss-Legendre Quadrature Based Expected Value Simulation for Optimizing Standby Redundancy System under Uncertain Environment"],"prefix":"10.1142","volume":"34","author":[{"given":"Yujie","family":"Gu","sequence":"first","affiliation":[{"name":"School of Business Administration, Henan Polytechnic University, Shiji Avenue 2001, Jiaozuo, 454000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qing","family":"Ma","sequence":"additional","affiliation":[{"name":"School of Business Administration, Henan Polytechnic 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